Microscopic entropy of de Sitter spacetime and entropic solution to the old cosmological constant problem
We study the role of Weyl symmetry breaking in conformal gravity and the residual scale symmetry of Einstein gravity. The corresponding action is characterized by a dimensionless coupling $α$, determined by the ratio between the de Sitter and Planck scales. We show that this quantity admits a natural interpretation as the Bekenstein-Hawking entropy of de Sitter spacetime. Combining ideas from the functional renormalization group, holography, and emergent gravity, we propose a microscopic interpretation of $α$ as a measure of the degrees of freedom associated with the de Sitter horizon. In this framework, the renormalization group flow of $α(k)$ encodes the scale dependence of these microscopic degrees of freedom. Requiring this flow to be monotonically increasing toward the infrared leads to a cosmological constant of the same order as the observed one, suggesting an entropic solution to the old cosmological constant problem. This remarkably small value can therefore be understood as a direct consequence of the extraordinarily large number of degrees of freedom in our de Sitter universe.